[EM] Yes, the Approval loser can be the CW.

[Forest Simmons]

Date: Sat, 12 Mar 2005 16:43:21 -0800
From: Russ Paielli <6049awj02 at sneakemail.com>
Subject: Re: [EM] Re: Chain Climbing --> Chain Filling
To: election-methods at electorama.com
Message-ID: <42338CA9.30501 at sneakemail.com>
Content-Type: text/plain; charset=ISO-8859-1; format=flowed

Ted Stern tedstern-at-mailinator.com |EMlist| wrote:

Is it possible for the least-approved candidate to be the Condorcet winner?


I reply, Yes:

Here's the example that has inspired so much ingenuity:

49 C>>A=B
24 B>>A>C
27 A>B>>C

Candidate A is the CW, but has the least approval, only 27, compared to 49 
for C and 51 for B.

Perhaps it is this weakness that makes A vulnerable to B's offensive 
truncation:

49 C
24 B
27 A>B

Jobst and I are convinced that B cannot confidently do this insincere 
truncation in the face of uncertainty introduced by a certain amount of 
randomization.

In my opinion, one key is to give both the CW and the Approval Winner (AW) 
some of the probability.  Part of Jobst's approach is to perturb the 
approval order through randomization so that B cannot count on all of the 
benefits of being the approval winner.  How best to combine these ideas in 
general is the subject of our quest.

My current tentative solution is to give every candidate that is not 
defeated pairwise by any other candidate with greater (randomly perturbed) 
approval a piece of the probability pie by random ballot.

In cases where candidate A is both the solid CW and the solid AW, no other 
candidate qualifies for a piece of the pie.

By "solid" I mean, "not easily perturbed."

The easier they are perturbed, the more the probability should be spread 
around.


Forest